On certain semigroups of finite monotone and order-decreasing partial transformations

Let ${\mathcal{P}\mathcal{M}\mathcal{D}}_n$ be the semigroup consisting of all monotone and order-decreasing partial transformations, and let ${\mathcal{I}\mathcal{M}\mathcal{D}}_n$ be the subsemigroup of ${\mathcal{P}\mathcal{M}\mathcal{D}}_n$ consisting of all injective monotone and order-decreasing transformations on the finite chain $X_n=\{1<\cdots <n\}$. For $2\le r\le n$, let $\mathcal{P}\mathcal{M}\mathcal{D}(n,r)=\{\alpha \in {\mathcal{P}\mathcal{M}\mathcal{D}}_n:\mathrm{\mid }\text{im}(\alpha )\mathrm{\mid }\le r\}$ and $\mathcal{I}\mathcal{M}\mathcal{D}(n,r)=\{\alpha \in {\mathcal{I}\mathcal{M}\mathcal{D}}_n:\mathrm{\mid }\text{im}(\alpha )\mathrm{\mid }\le r\}$. In this paper, we determine the cardinalities, maximal subsemigroups and ranks of $\mathcal{P}\mathcal{M}\mathcal{D}(n,r)$ and $\mathcal{I}\mathcal{M}\mathcal{D}(n,r)$, and moreover, we verify that the semigroups $\mathcal{P}\mathcal{M}\mathcal{D}(n,r)$ and $\mathcal{I}\mathcal{M}\mathcal{D}(n,r)$ are non-regular but abundant for any $2\le r\le n$.